The Importance of the Concept of Equivalence in Children Learning Mathematics

Any figure, step, numerical look, algebraic look, or equation can be represented in an infinite figure of ways that have the same value ( Charles, 2005, p.10 )

Forming equalities ( or recognizing similarities ) is one of the most of import cognitive procedures we use to organize and do sense of all our experiences ( Haylock, 2010, p.27 ) . In mathematics, an equality is formed when two or more mathematical entities are the same or portion belongingss.

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Harmonizing to Haylock and Thangata ( 2007, p.62 ) the construct of equality is cardinal in the experience of kids larning mathematics, both in footings of conceptual acquisition and as a tool for pull stringsing mathematical thoughts. Indeed Charles ( 2005, p.10 ) claims that equality is a ‘Big Idea’ as it is an thought that is cardinal to the acquisition of mathematics, i.e. it links legion mathematics apprehensions into a consistent whole.

Large thoughts make connexions, an of import constituent which contributes to ‘Mathematical Literacy’ , a term coined by PISA ( 2006 ) , who believe that doing connexions between thoughts enables kids to utilize mathematics in real-life contexts, therefore guaranting mathematical apprehension, which in bend contributes to understanding the large thought ( ACME,2008, p.10 ) .

Research by ACME ( 2006, p.2 ) has found that some kids become dying about mathematics due to a deficiency of a solid apprehension of the topic. The course of study needs to be process-driven or conceptual instead than procedural. A position shared by Skemp ( 1976 ) . If pupils experience a carefully developed patterned advance of mathematical constructs and thoughts they will develop as mathematical minds. Connections should be made between mathematics and other course of study countries.

It is seldom possible to take cutoffs to larning complex facets of mathematics without first geting on the manner more simple thoughts ( ACME, 2008, p.7 ) .

Equivalence takes many signifiers in mathematics, get downing in the really early old ages of instruction when for illustration, kids recognise that although a set of three toads is different from three teddies, they both portion the belongings of ‘threeness’ . This can be shown by one-to-one numeration and, moreover, can be abstracted from other aggregations of ‘three’ , to organize the construct of ‘three’ as a central figure, independent of any specific context.

Subsequently on, kids learn that Numberss can be decomposed into parts in an infinite figure of ways. Understanding that representations can be change while keeping the same value or the same solution is a powerful problem-solving tool. For illustration, 5 + 6 is the same as 5 + 5 and 1 more, and 6 ten 7 is the same as 5 x 7 and 7 more. Both of these schemes, and others, are connected through the thought of equality ; both involve interrupting the computation apart into an tantamount representation that uses known facts to calculate out the unknown fact ( Charles, 2010, p.10 ) .

Eloquence in the country of fractions is considered indispensable for accessing the secondary mathematics course of study, peculiarly step, algebra, geometry and chance. However, “children’s development of both conceptual and procedural cognition of fractions has been identified as a cardinal difficulty” ( DfE, 2012, p.73 ) . Equality is peculiarly of import in this country as kids need to understand that every fraction can be represented by an infinite set of different bur tantamount fractions in order to entree fractional jobs affecting ordination, adding and deducting fractions ( DfES, 2002, p.10 ) .

Without an apprehension of tantamount fractions students will fight to compare rational Numberss and execute operations with fractions ( NCETM, 2014, lines 1-2 ) .

An apprehension of equivalency from the early old ages of instruction is besides important to the apprehension of algebraic constructs. By learning kids the accomplishment of generalization in arithmetic by usage of equality, and how to show that generalization utilizing progressively formal linguistic communications they will understand that to understand algebra is to do a rich web of connexions throughout the full mathematics course of study. For illustration, a pupil comparing different monetary values might generalize that a little addition in monetary value will non alter the result of the comparing. This difference of context can do mathematics involved more of import to the pupil both in the short and long term ( Kaput, 2000, p.8 ) . A steadfast appreciation of equality as a symmetric relation ( a=b hence b=a ) will intend that sense can be given to a statement like 5=3+2, which is frequently rejected by kids. Since a=b and b=c implies that a=c, equality is besides transitive, and automatic ( a=a ) ( ACME ( 2008, p.5 )

These thoughts which relate to Numberss and figure operations can frequently be extended to other countries of mathematics at all phases of acquisition. For illustration, symmetricalness and steps, in understanding that a fixed measure of liquid occupies the same volume irrespective of the container it is in. Or that the fixed mass of an object remains unchanged irrespective of its form. Or that a one-fourth of an hr is the same sum of clip whether it is from 3pm to 3.15pm or from 3.10pm to 3.25pm etc. Similarly, equality dealingss can be applied to determine, infinite and steps. “Understanding equality will enable students to exchange easy between units of measures” ( ACME, 2008, p.5 ) .

The instructor needs to hold deep topic cognition in order to assist kids do connexions between different parts of the mathematics curriculum throughout every lesson, irrespective of its chief subject ( p.44 ) ( Stacey, 2006 ) .

Good instruction should do these thoughts explicit. Large thoughts are of import for kids. Large thoughts must be taught explicitly. Children must non be expected to see the connexions independently ( Charles, 2005, p.10 ) .

Equality unifies different facets of mathematics and demonstrates how they are related. It is every bit much about weights and steps as it is about fractions. It is every bit much about Pythagoras’ Theorem as it is about the commutative jurisprudence of generation or relative logical thinking. It offers deep conceptual apprehension of how we can utilize premier factors to demo that 6 ten 8 is tantamount to 3 tens 16, as we have halved one of the Numberss and duplicate the other. The existent solution is non every bit of import as the understanding it involves. Equality is a cardinal construct in most countries of mathematics since set uping equality between two constructs allows each to go portion of a higher order theory.

Understanding of math equality involves several related accomplishments:

• Interpreting the equal mark as relational symbol

• Reading and encoding the construction of equations right

• Noticing dealingss among looks in equations

• Correctly placing and equalising the two sides of an equation

( p.1 )

The general premise is that a better apprehension of mathematical equality in the early classs leads to greater success in mathematics as kids progress through school, into algebra, and beyond ( McNeil, 2013, p.15 ) .

The activity detailed in portion two of this paper will show how we can assist kids develop understanding in these countries.

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